Mathematics for the interested outsider

Consequences of the Submodule Theorem

We have a number of immediate consequences of the submodule theorem. First, and most important, the Specht modules form a complete list of irreducible modules for the symmetric group . We know that they’re irreducible, and that there’s one of them for each partition , which is the number of modules we’re looking for. But we need to show that the Specht modules corresponding to distinct partitions are themselves distinct. For this, we’ll use a lemma.

If is a nonzero intertwinor, then . Further, if , then must be multiplication by a scalar. Indeed, since there must be some polytabloid with . We decompose , and extent to all of by sending every vector in to . That is:

where the are -tableaux. Now, the can’t all be zero, so we must have at least one -tableau so that . But then our corollary of the sign lemma tells us that , as we asserted!

Further, if , then our other corollary shows us that for some scalar . We can thus calculate

and so multiplies every vector by .

As a consequence, the must be distinct for distinct permutations, since if then there is a nonzero homomorphism , and thus . But the same argument shows that , and thus .

More particularly, we have a decomposition

where the diagonal multiplicities are . The rest of these multiplicities will eventually have a nice interpretation.

About this weblog

This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.